L27 ARFIMA Processes
Autoregressive Fractionally Integrated Moving Average Process
- Persistence and Anti-persistence (mean-reverting) processes
- Integrated ARMA models reduce to stationary after short range dependence after a finite number of differences.
- ARMA cannot capture long term dependence (persistence)
- Whereas ARFIMA ACF decays slowly.
Fractionally Integrated Noise¶
\(ARFIMA(0,d,,0)\) process.
\(Y_{t}\) is fractionally integrated noise or fractionally integrated ARMA process of the order \((0,d,0)\) with \(-0.5 \lt d \lt 0.5\).
\(y_{t}\) is a stationary solution of \((1-B)^dy_{t} = e_{t}\) where \(e_{t}\) is white noise (with variance \(\sigma_{e}^{2}\))
Thus,
\[
y_{t} = (1-B)^{-d} e_{t}
\]
What values of \(d\) will it have a long lasting impact?
Redefine,
\[
(1-B)^{-d} = \sum_{j=0}^\infty \psi_{j}B^j
\]
where, \(\psi_{j} = \dfrac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}\). And, \(\Gamma(\alpha) = (\alpha-1)!\)
If \((-0.5 \lt d \lt 0.5)\),
\[
y_{t} = \sum_{j=0}^\infty \psi_{j} e_{t-j}
\]
Use Stirling's approximation for \(\Gamma(x)\) as \(x\to\infty\)
- If \(d \sim 0.5\), \(\rho_{k}\) decays very slowly → Long Memory
- If \(d \sim -0.5\), \(\rho_{k}\) decays rapidly → short memory
- \(d\lt 0\) \(\implies\) Process is an intermediate memory process
- \(0 \lt d\lt 0.5\) \(\implies\) Process is an long memory process
ARFIMA model structure
\[
(1-B)^d Y_{t} = \phi(B)Y_{t} + \theta(B)e_{t}
\]
- \(d=0\) \(\implies\) ARMA model
- \(d=1\) \(\implies\) ARIMA model